Solve for x using the quadratic formula. [tex]x^2-12 x+61=0[/tex]
Answer (1)
Identify the coefficients: a = 1 , b = − 12 , c = 61 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 2 12 ± − 100 = 2 12 ± 10 i .
Obtain the complex solutions: x = 6 ± 5 i . Thus, the solutions are 6 + 5 i and 6 − 5 i .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 12 x + 61 = 0 and asked to solve for x using the quadratic formula.
Stating the Quadratic Formula The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 .
Identifying Coefficients In our equation, x 2 − 12 x + 61 = 0 , we can identify the coefficients as a = 1 , b = − 12 , and c = 61 .
Substituting into the Formula Now, we substitute these values into the quadratic formula: x = 2 ( 1 ) − ( − 12 ) ± ( − 12 ) 2 − 4 ( 1 ) ( 61 )
Simplifying the Expression Simplify the expression: x = 2 12 ± 144 − 244 x = 2 12 ± − 100
Dealing with the Negative Discriminant Since the discriminant (the value inside the square root) is negative, we will have complex solutions. We know that − 1 = i , so − 100 = 10 i .
Substituting the Imaginary Unit Substitute this back into the equation: x = 2 12 ± 10 i
Simplifying to Final Solutions Finally, divide both terms in the numerator by 2: x = 6 ± 5 i
Final Answer Therefore, the solutions for x are x = 6 + 5 i and x = 6 − 5 i .
Examples
Quadratic equations are not just abstract math; they appear in physics when calculating projectile motion, in engineering for designing structures, and even in economics for modeling cost functions. For example, if you're launching a rocket, the height of the rocket over time can be modeled by a quadratic equation. Solving this equation tells you when the rocket will hit the ground or reach a certain altitude. Understanding quadratic equations helps engineers and scientists make accurate predictions and build safe and efficient systems.
